I want to apply a spectral method for the weak formulation of the equation
$(-\Delta)^su=f$ $s>0$
with zero Dirichlet boundary conditions, where $(-\Delta)^s$ shall be the fractional laplacian on an intervall of the form $(a,b)$.
I assume I have a ONB in $L^2(a,b)$ $\phi_i$ that is an ONS $H^s_0(a,b)$.
Then the weak formulation reads, when $u_N:=\sum\limits_{i=1}^N \alpha_i\varphi_i$ is the projection of $u$ on $L^2$
$\lambda_i \alpha_i =(f, \varphi_i)_{L^2}$ $(i=1,\ldots, n)$
What basis functions are available whose regularity explicitly depend on the $s$ i.e. only differentiable up to the order $s$ not higher.